Unbounded expectations to some von Neumann algebras

TL;DR

This paper constructs a special type of expectation map for certain von Neumann algebras, extending classical concepts to unbounded settings and highlighting differences based on group properties.

Contribution

It introduces unbounded expectations for injective von Neumann algebras under group actions, generalizing the notion of conditional expectations.

Findings

Unbounded expectations exist for injective von Neumann algebras with group actions.

For amenable groups, the unbounded expectation reduces to a norm-1 conditional expectation.

The constructed maps are R-bimodular and preserve positivity relations.

Abstract

For any injective von Neumann algebra R and any discrete, countable group G, which acts by *-automorphisms on R, we construct an idempotent mapping of an ultra-weakly dense subspace of B(H) onto the reducerd crossed product von Neumann algebra, such that it is R-bimodular and satisfies some nice relations with respect to positivity. In the case of an amenable group our unbounded expectation turns into a usual conditional expectation of norm 1.